The "Zeno's Paradoxes and Calculus" Paradox

One of my fondest memories of taking freshman calculus was the brief discussion of Zeno's paradoxes. For anyone unfamiliar, the particular one I remember is the Dichotomy paradox:

That which is in locomotion must arrive at the half-way stage before it arrives at the goal.

--Aristotle, Physics VI:9, 239b10

That is, if you are walking towards the wall, first you must travel half way there.  Then you must travel half way from there to the wall, then half way again, and so on.  Thus you will never reach the wall!  (Actually, this is backwards from the traditional reading of the paradox: before you travel half way there, you must first travel half way to that half way point, and before that, half way to there, so in fact you never start moving at all!)

This paradox is often used as an example of a great mystery that calculus can help us solve.  I so wish that were the case.

I have given my students this example in my own calculus courses.  It always goes over very well.  When first describing the situation, I ask given all this whether I will ever reach the wall, to which most students say that I will not.  "Great," I say, and proceed to walk straight into the wall.  It is a fun activity that engages students, and is related to mathematics.  Except that it is not related to mathematics.

The usual explanation of the paradoxes using calculus is to show that the geometric series with ration 1/2 converges.  But this is not what is perplexing about Zeno's paradox.  In fact, doing the math behind this series is much more complicated than just looking at a picture of a line divided first in half, then the next part in half again, and so on.  Clearly the sum of 1/2, 1/4, 1/8, ... is 1.  The fact that mathematicians have been able to develop a the notions of limit and infinite series to a level of precision which agrees with our intuition is remarkable, yes, but the result is not surprising.  This explanation acts as if the perplexing thing about Zeno's paradox is that the result of traveling these half distances is just the whole distance, and not an infinite distance.  After all, at first glance, adding up an infinite number of things should not give you something finite.

Perhaps better would be to use the geometric series to represent time.  Say you walk one mile at one mile per hour.  In half an hour, you have walked half a mile.  Then you need 1/4 of an hour to get through the next 1/4 of a mile.  Then you need 1/8 of an hour to go the next bit, then 1/16 of an hour, and so on.  You add up all these times, you get 1 hour, and you have traveled 1 mile (again by adding up all those distances).  Alright, so this definitely is convincing.  I am now sure that I will reach my goal in a finite amount of time.  Of course, I knew that already: I walked right into the wall.  Anyway, Aristotle even gave that explanation, and he didn't know any calculus.

The reason Zeno's paradox is compelling is that it requires you accomplish an infinite number of steps.  Not an infinite number of steps in 1 hour, but an infinite number of steps at all.  I can see that this particular infinite sum is a finite number, but what I cannot see is that I would be able to ever arrive at that number by physically entering infinitely many terms into my calculator (even if I could do so at an ever increasing rate).  This seems like a problem for physics, not mathematics. 

And yet, Zeno's paradox is such a great teaching tool.  If only there were a way to use it that did it justice.